Calculus 6.1

1
6.1 Antiderivatives and Slope Fields
Greg Kelly, Hanford High School, Richland,
Washington
2
First, a little review
It doesnt matter whether the constant was 3 or
-5, since when we take the derivative the
constant disappears.
However, when we try to reverse the operation
We dont know what the constant is, so we put C
in the answer to remind us that there might have
been a constant.
3
If we have some more information we can find C.
4
Initial value problems and differential equations
can be illustrated with a slope field.
5
0
0
0
0
1
0
0
0
2
0
0
3
2
1
0
1
1
2
2
0
4
-1
-2
0
0
-4
-2
6
If you know an initial condition, such as (1,-2),
you can sketch the curve.
By following the slope field, you get a rough
picture of what the curve looks like.
In this case, it is a parabola.
7
For more challenging differential equations, we
will use the calculator to draw the slope field.
(Notice that we have to replace x with t , and y
with y1.)
(Leave yi1 blank.)
8
Set the viewing window
Then draw the graph
9
Be sure to change the Graph type back to FUNCTION
when you are done graphing slope fields.
10
(No Transcript)
11
Integrals such as are called
indefinite integrals because we can not find a
definite value for the answer.
12
Many of the integral formulas are listed on page
307. The first ones that we will be using are
just the derivative formulas in reverse.
On page 308, the book shows a technique to graph
the integral of a function using the numerical
integration function of the calculator (NINT).
This is extremely slow and usually not worth the
trouble.
A better way is to use the calculator to find the
indefinite integral and plot the resulting
expression.
13
To find the indefinite integral on the TI-89, use
The calculator will return
Notice that it leaves out the C.
14
p